Fano fibrations and DK conjecture for relative Grassmann flips

Given a vector bundle $\mathcal E$ on a smooth projective variety $B$, the flag bundle $\mathcal F l(1,2,\mathcal E)$ admits two projective bundle structures over the Grassmann bundles $\mathcal G r(1, \mathcal E)$ and $G r(2, \mathcal E)$. The data of a general section of a suitably defined line bundle on $\mathcal F l(1,2,\mathcal E)$ defines two varieties: a cover $X_1$ of $B$ and a fibration $X_2$ on $B$ with general fiber isomorphic to a smooth Fano variety. We construct a semiorthogonal decomposition of the derived category of $X_2$ which consists of a list of exceptional objects and a subcategory equivalent to the derived category of $X_1$. As a byproduct, we obtain a new full exceptional collection for the Fano fourfold of degree $12$ and genus $7$. Any birational map of smooth projective varieties which is resolved by blowups with exceptional divisor $\mathcal F l(1, 2, \mathcal E)$ is an instance of a so-called Grassmann flip: we prove that the DK conjecture of Bondal-Orlov and Kawamata holds for such flips. This generalizes a previous result of Leung and Xie to a relative setting.